Related papers: Best rational approximations of an irrational numb…
We begin a discussion about the maximal containments of lower central series ideals: ideals generated by products of two-sided ideals of the lower central series of the free associative algebra on $n$ generators. We introduce two new ideas…
We study the reverse mathematics of pigeonhole principles for finite powers of the ordinal $\omega$. Four natural formulations are presented and their relative strengths are compared. In the analysis of the pigeonhole principle for…
Introducing the notion of a rational system of measure preserving transformations and proving a recurrence result for such systems, we give sufficient conditions in order a subset of rational numbers to contain arbitrary long arithmetic…
We introduce algorithms that use predictions from machine learning applied to the input to circumvent worst-case analysis. We aim for algorithms that have near optimal performance when these predictions are good, but recover the…
Number sequences defined by a linear recursion relation are studied by means of generating functions. Indices of the terms in the recursion relation have arbitrary differenses. In addition to formulas for the nth term an algorithm is…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
New algorithms are presented for numerical conformal mapping based on rational approximations and the solution of Dirichlet problems by least-squares fitting on the boundary. The methods are targeted at regions with corners, where the…
Let $\al$ be an irrational and $\varphi: \N \rightarrow \R^+$ be a function decreasing to zero. For any $\al$ with a given Diophantine type, we show some sharp estimations for the Hausdorff dimension of the set [E_{\varphi}(\al):={y\in \R:…
I formulate the problem of closing the detection loophole as a constrained optimization problem. Numerical methods can then be used to maximize the detector efficiency subject to the constraint that there exists a local realist explanation…
In this notes we make a comparison between the arithmetic properties of irrational numbers and their dynamical properties under the Gauss map. We show some equivalences between different classifications of irrational numbers such as the…
Considering simultaneous approximation to three numbers, we study the geometry of the sequence of best approximations. We provide a sharper lower bound for the ratio between ordinary and uniform exponent of Diophantine approximation,…
Fix an irrational number $\theta$. For a real number $\tau >0$, consider the numbers $y$ satisfying that for all large number $Q$, there exists an integer $1\leq n\leq Q$, such that $\|n\theta-y\|<Q^{-\tau}$, where $\|\cdot\|$ is the…
We discuss some examples that illustrate the countability of the positive rational numbers and related sets. Techniques include radix representations, Godel numbering, the fundamental theorem of arithmetic, continued fractions, Egyptian…
We axiomatize the first-order theories of exponential integer parts of real-closed exponential fields in a language with $2^x$, in a language with a predicate for powers of 2, and in the basic language of ordered rings. In particular, the…
The notions of fractal and essentially fractal algebras of approximation sequences and of the Arveson dichotomy have proved extremely useful for several spectral approximation problems. The purpose of this short note is threefold: to…
The multiplicative theory of a set of numbers (which could be natural, integer, rational, real or complex numbers) is the first-order theory of the structure of that set with (solely) the multiplication operation (that set is taken to be…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
This paper proves the approximate intermediate value theorem, constructively and from notably weak hypotheses: from pointwise rather than uniform continuity, without assuming that reals are presented with rational approximants, and without…
The optimal mass transport problem gives a geometric framework for optimal allocation, and has recently gained significant interest in application areas such as signal processing, image processing, and computer vision. Even though it can be…
We consider the problem of computing the nearest matrix polynomial with a non-trivial Smith Normal Form. We show that computing the Smith form of a matrix polynomial is amenable to numeric computation as an optimization problem.…